The Number of 0-1-2 Increasing Trees as Two Different Evaluations of the Tutte Polynomial of a Complete Graph - Merino, C. (The Electronic Journal of Combinatorics, 2008)
If T_n(x,y) is the Tutte polynomial of the complete graph K_n, we have the equality T_n+1(1,0)=T_n(2,0). This has an almost trivial proof with the right combinatorial interpretation of T_n(1,0) and T_n(2,0). We present an algebraic proof of a result with the same flavour as the latter: T_n+2(1,-1)=T_n(2,-1), where T_n(1,-1) has the combinatorial interpretation of being the number of 0–1–2 increasing trees on n vertices.